id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-a4ae81b53026fc043ae38d2a5d2edb6e20bd5df7f5dd135ca0673595fb35b976 | What is the value of $n$ such that both $4n+1$ and $9n+1$ are perfect squares? | [
"0"
] | [
"What is the value of $n$ such that both $4n+1$ and $9n+1$ are perfect squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6a491cbacfbbbcdf2584cddd57002626b324c009e1398eb9a68742e8f30e3046 | What is the value of $s$ in the equation $1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 = s^2$? | [
"36"
] | [
"What is the value of $s$ in the equation $1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 = s^2$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5821f73cc11370a3230630ea6f78cd0f40e6ead5cb9679a1041c70a7be596f09 | What is the value of $x + y$ if the sequence $2, ~6, ~10, \ldots, ~x, ~y, ~26$ is an arithmetic sequence? | [
"40"
] | [
"What is the value of $x + y$ if the sequence $2, ~6, ~10, \\ldots, ~x, ~y, ~26$ is an arithmetic sequence?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-03820421a9017abec17aa23f0f657c759374a0a104be4877f1284c0cb14ca803 | What is the value of $x$ for which $(2008+x)^2=x^2$? | [
"-1004"
] | [
"What is the value of $x$ for which $(2008+x)^2=x^2$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d9b4c5aa9cac5d35c99c07b4eccc755c9e3768e28a95f3760ca29610185e9e5 | What is the value of $x$ for which $(8-x)^2=x^2$? | [
"4"
] | [
"What is the value of $x$ for which $(8-x)^2=x^2$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2bf7d2bc90d30113377c081dbb3f56bac4878adb7746802afa97cbcb581d0463 | What is the value of $x$ for which $|3x+5|$ is not positive? Express your answer as a common fraction. | [
"-\\frac{5}{3}"
] | [
"What is the value of $x$ for which $|3x+5|$ is not positive? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5ba22f6f85b11e5ce3d1bf028902191976dbf13811433b6b0698e6c53f54fa32 | What is the value of $x$ if $x=\frac{2009^2-2009}{2009}$? | [
"2008"
] | [
"What is the value of $x$ if $x=\\frac{2009^2-2009}{2009}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2c148febd20a984b6fc67bbd2f55e7eac7dd819c530519d3e8ea5ebe64bd3acc | What is the value of $x$ if $|x-1| = |x-2|$? Express your answer as a common fraction. | [
"\\frac{3}{2}"
] | [
"What is the value of $x$ if $|x-1| = |x-2|$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cbdd42f4e3dd315d1025ae48fee3e1bf7594669936e0bfad1d75f589aa679d3c | What is the value of $x$ if \begin{align*}x &= y+5,\\ y &= z+10,\\ z &= w+20,\\ \text{and }\qquad w &= 80? \end{align*} | [
"115"
] | [
"What is the value of $x$ if \\begin{align*}x &= y+5,\\\\\ny &= z+10,\\\\\nz &= w+20,\\\\\n\\text{and }\\qquad w &= 80?\n\\end{align*}"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b29a8929da4e592520d360a6fce80f341025b61f865775616b550b681320c904 | What is the value of $x$ if a cube's volume is $5x$ cubic units and its surface area is $x$ square units? | [
"5400"
] | [
"What is the value of $x$ if a cube's volume is $5x$ cubic units and its surface area is $x$ square units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1fdb37e512b1fb8942f6e0af360228a71acf1502bb121f804bdb0ff673241e31 | What is the value of $x$ in the equation $16^{16}+16^{16}+16^{16}+16^{16}=2^x$? | [
"66"
] | [
"What is the value of $x$ in the equation $16^{16}+16^{16}+16^{16}+16^{16}=2^x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-477a46e5e4b752a202edd15d8e3c2c751fec82de6df99f1ead87a791920224a2 | What is the value of $x$ in the equation $9^4+9^4+9^4=3^x$? | [
"9"
] | [
"What is the value of $x$ in the equation $9^4+9^4+9^4=3^x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7647cd4c007c72099a9a7a5576ed4491e1d5dfb5af2f44e4578c8c6a3a3981bc | What is the value of $x$ in the equation $\frac{1}{x} + \frac{2}{x} \div \frac{4}{x} = 0.75$? | [
"4"
] | [
"What is the value of $x$ in the equation $\\frac{1}{x} + \\frac{2}{x} \\div \\frac{4}{x} = 0.75$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb5643941ac079c793bc3b8d62f729332477feb380ec49dfa1f42d2299a02530 | What is the value of $x$ so that the line passing through $(x,5)$ and $(4,11)$ has a slope of $6$? | [
"3"
] | [
"What is the value of $x$ so that the line passing through $(x,5)$ and $(4,11)$ has a slope of $6$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-30ac13d2d9b4aeadea075cdda69b2391756873bdbdea2289c057f2fea01e4b51 | What is the value of $x$ that satisfies $\log_4 (\log_2 x) + \log_2 (\log_4 x) = 2$? | [
"16"
] | [
"What is the value of $x$ that satisfies $\\log_4 (\\log_2 x) + \\log_2 (\\log_4 x) = 2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc46efb7fddb7e852a63e897c2db35c6edd6c71971399c54023be5cdf864af71 | What is the value of $x$ when $\cos(x) = x$? | [
"0.739085"
] | [
"What is the value of $x$ when $\\cos(x) = x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50756b199d70e1b8659e15b7a571a3f84594e9cbd3ae0b70797972f513fde6c1 | What is the value of $x$ when $\sin(2x) = -\frac{8}{9}$? | [
"\\( x = \\frac{1}{2}(2\\pi n - \\sin^{-1}(\\frac{8}{9})), \\quad n \\in \\mathbb{Z} \\) or \\( x = \\frac{1}{2}(2\\pi n + \\pi + \\sin^{-1}(\\frac{8}{9})), \\quad n \\in \\mathbb{Z} \\)"
] | [
"What is the value of $x$ when $\\sin(2x) = -\\frac{8}{9}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 24
} | 37 | 0 | 0 |
omr-ws-b158dad9e92c225d566bf3684cdba796812614fb157f60c2d1484e9f0f6fd805 | What is the value of $x^2+y^2-z^2+2xy$ when $x=-3$, $y=5$, and $z=-4$? | [
"-12"
] | [
"What is the value of $x^2+y^2-z^2+2xy$ when $x=-3$, $y=5$, and $z=-4$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-45de2aa6f695e30856947fb521f57e05cee2eda5a7b900b6aecc09d3f213556a | What is the value of $x^5 + 8x^3 - 6x^2 - 9x + 8$ if $x = \sqrt[3]{9} - \sqrt[3]{3}$? | [
"2"
] | [
"What is the value of $x^5 + 8x^3 - 6x^2 - 9x + 8$ if $x = \\sqrt[3]{9} - \\sqrt[3]{3}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a67f01185e54486ec6ff549e5334eabf40e034009239b12f99b7dbe3865fa4c1 | What is the value of $x^5 - 5x$ when $x = 4$? | [
"1004"
] | [
"What is the value of $x^5 - 5x$ when $x = 4$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b683f38dba0a7f02af524fbc229a84f5096a8b2196968f311a06a6fd83c8c14 | What is the value of $y$ in the equation $\frac{30}{50}=\sqrt{\frac{y}{50}}$? | [
"18"
] | [
"What is the value of $y$ in the equation $\\frac{30}{50}=\\sqrt{\\frac{y}{50}}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42d83cc50ff84ecd20e4d47366d184dbdc051f5449d91ad6f7ac2e673e7b2e76 | What is the value of ${0 \choose 1/2}$? | [
"\\(\\frac{2}{\\pi}\\)"
] | [
"What is the value of ${0 \\choose 1/2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 30
} | 48 | 0 | 0 |
omr-ws-0488d76442cf3bf38f76aa0413def67aa95097dc1cc5170b76f6c898d1ec9bac | What is the value of ${{{{{{{{i^{i}}^{i}}^{i}}^{.}}^{.}}^{.}}^{i}}^{i}}$, where $i$ occurs $i$ times? | [
"0.438 + 0.361i"
] | [
"What is the value of ${{{{{{{{i^{i}}^{i}}^{i}}^{.}}^{.}}^{.}}^{i}}^{i}}$, where $i$ occurs $i$ times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-57e74425a62b859add1924a8ea1dcc5cf39f642f9a51e0534b7d4c55eeb2abcb | What is the value of $|(w_1,w_2)/(aw_1+bw_2,cw_1+dw_2)|$, where $(w_1,w_2)=w_1\mathbb{Z}+w_2\mathbb{Z}$ and $w_1,w_2\in\mathbb{C}$? | [
"|ad - bc|"
] | [
"What is the value of $|(w_1,w_2)/(aw_1+bw_2,cw_1+dw_2)|$, where $(w_1,w_2)=w_1\\mathbb{Z}+w_2\\mathbb{Z}$ and $w_1,w_2\\in\\mathbb{C}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-51ecc4c1a8391fc17954e47bd38d36d6a6e551babb2bdee165f9ab02af8241b5 | What is the value of \( A \) when: \[ A = \sqrt{17 - \sqrt{288}} + \sqrt{15 - \sqrt{224}} + \sqrt{13 - \sqrt{168}} + \sqrt{11 - \sqrt{120}} + \sqrt{9 - \sqrt{80}} \] | [
"1"
] | [
"What is the value of \\( A \\) when:\n\\[ A = \\sqrt{17 - \\sqrt{288}} + \\sqrt{15 - \\sqrt{224}} + \\sqrt{13 - \\sqrt{168}} + \\sqrt{11 - \\sqrt{120}} + \\sqrt{9 - \\sqrt{80}} \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-083a6e1787e71f5b0b3eee19a22856c4d757445b5e2520c869c8668787ac475c | What is the value of \( E = \tan\left(\frac{\pi}{9}\right) + \tan\left(\frac{4\pi}{9}\right) + \tan\left(\frac{7\pi}{9}\right) \)? | [
"\\( 3\\sqrt{3} \\)"
] | [
"What is the value of \\( E = \\tan\\left(\\frac{\\pi}{9}\\right) + \\tan\\left(\\frac{4\\pi}{9}\\right) + \\tan\\left(\\frac{7\\pi}{9}\\right) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3a8552ddb9bde0ea4fdec9d912b45a8dd0ae8df6388838006ac6bea9f9fbed35 | What is the value of \( \binom{2000}{2} + \binom{2000}{5} + \binom{2000}{8} + \cdots + \binom{2000}{2000} \)? | [
"\\(\\frac{2^{2000} - 1}{3}\\)"
] | [
"What is the value of \\( \\binom{2000}{2} + \\binom{2000}{5} + \\binom{2000}{8} + \\cdots + \\binom{2000}{2000} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-390198381c8c189139c8054678567667384d9877ae152a8de0af5390ffa4cade | What is the value of \( \cos \left(\frac{\pi}{9}\right) \cos \left(\frac{4\pi}{9}\right) \cos \left(\frac{7\pi}{9}\right) \)? | [
"\\(-\\frac{1}{8}\\)"
] | [
"What is the value of \\( \\cos \\left(\\frac{\\pi}{9}\\right) \\cos \\left(\\frac{4\\pi}{9}\\right) \\cos \\left(\\frac{7\\pi}{9}\\right) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-cd3b4e829561552633eceb79e073c6f7c080beb8e3e97810bacf5184c9825820 | What is the value of \( \frac{1/2}{3/4} + \frac{5/6}{7/8} + \frac{9/10}{11/12} + \cdots + \frac{97/98}{99/100} \)? | [
"\\dfrac{1175}{51}"
] | [
"What is the value of \\( \\frac{1/2}{3/4} + \\frac{5/6}{7/8} + \\frac{9/10}{11/12} + \\cdots + \\frac{97/98}{99/100} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1200bbf4ed00079b494a8ad097bd8510e4c213e890818568a5aa96ea71afc16 | What is the value of \( \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} e^{\frac{k}{n}} \)? | [
"e - 1"
] | [
"What is the value of \\( \\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{k=1}^{n} e^{\\frac{k}{n}} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-89b3b6b2694e1a132d87290b28d6d798bf97da903efb6ece848c3bd4d797262f | What is the value of \( \lim_{n \to \infty} \left( |A_n \cap A_{n+1}| \right) \), where \( A_n = \{n, n+1, n+2, \ldots\} \)? | [
"\\(\\infty\\)"
] | [
"What is the value of \\( \\lim_{n \\to \\infty} \\left( |A_n \\cap A_{n+1}| \\right) \\), where \\( A_n = \\{n, n+1, n+2, \\ldots\\} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-084b9f90873bc3c20949e18f3a879ba350e890e83f385f5c1d6f89c45fe9b4ed | What is the value of \( \lim_{x \to 0} \sin\left(\frac{1}{x}\right) \)? | [
"The limit does not exist."
] | [
"What is the value of \\( \\lim_{x \\to 0} \\sin\\left(\\frac{1}{x}\\right) \\)?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-2dbf9fb812022b7a3b8faf0691c88de2c5539bd6c9bd882bcf508120540567d3 | What is the value of \( \sum_{k=3}^{\infty} \sin^k\left(\frac{\pi}{6}\right) \)? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the value of \\( \\sum_{k=3}^{\\infty} \\sin^k\\left(\\frac{\\pi}{6}\\right) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-81ff2568ec0ca07985bbcb318e09c7093b1db81bc183b66d1e464687fdc8f8ba | What is the value of \( \sum_{n=0}^{\infty} \dfrac{4^{2^{n-1}}}{1-4^{2^{n}}} \)? | [
"-1."
] | [
"What is the value of \\( \\sum_{n=0}^{\\infty} \\dfrac{4^{2^{n-1}}}{1-4^{2^{n}}} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-03d78b699ff59177a050a2b000a8e647a99d3df812508204ec2a3da959953d2e | What is the value of \( \tan^2 1^\circ + \tan^2 3^\circ + \cdots + \tan^2 89^\circ \)? | [
"4005"
] | [
"What is the value of \\( \\tan^2 1^\\circ + \\tan^2 3^\\circ + \\cdots + \\tan^2 89^\\circ \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-8d71e148d821d309cb4ad43c7095a21419342601fb10b6054cfbd23c3aacb362 | What is the value of \( \tan^6 40^\circ - 33 \tan^4 40^\circ + 27 \tan^2 40^\circ \)? | [
"3"
] | [
"What is the value of \\( \\tan^6 40^\\circ - 33 \\tan^4 40^\\circ + 27 \\tan^2 40^\\circ \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e692e6c46b2054015d21d4d95aabe4bc5874fd814647f747c334604215691c8 | What is the value of \( \text{cot}[\text{Arc Sin}(\frac{-3}{\sqrt{13}})] \)? | [
"\\(\\frac{2}{3}\\) or \\(-\\frac{2}{3}\\)"
] | [
"What is the value of \\( \\text{cot}[\\text{Arc Sin}(\\frac{-3}{\\sqrt{13}})] \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-8af63a4265b095fce842fbb0dcf3450381475a899e0b964a862b13403cbd05ce | What is the value of \( a \) if the system of equations \[ x^2 + y^2 = z \] \[ x + y + z = a \] has only one solution? | [
"\\(-\\frac{1}{2}\\)"
] | [
"What is the value of \\( a \\) if the system of equations \n\\[ x^2 + y^2 = z \\]\n\\[ x + y + z = a \\]\nhas only one solution?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-754f06e38e6fcac0c6fd8ecf2455d1906f50002b40950e130d3c621ee70187e7 | What is the value of \( e^\pi \times \pi^e \)? | [
"519.72"
] | [
"What is the value of \\( e^\\pi \\times \\pi^e \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 31,
"genselect": 7
} | 38 | 0 | 0 |
omr-ws-581e3dbe43604e0f7c23b6fa086d6bc30523540145bbe884d0ee7a9cb182522a | What is the value of \( i^\pi \)? | [
"\\( i^\\pi = \\exp(i\\pi(\\frac{\\pi}{2} + 2n\\pi)) \\) for \\( n \\in \\mathbb{Z} \\)"
] | [
"What is the value of \\( i^\\pi \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-709a84dc45555c361d193ec00cfea8cdf56cf50d1c3ae13c7dd26affa3072d1a | What is the value of \( k \) such that the equation \( x^2 - kx + 4 = 0 \) has a repeated real solution? Provide the steps to find the value of \( k \). | [
"\\( \\pm 4 \\)"
] | [
"What is the value of \\( k \\) such that the equation \\( x^2 - kx + 4 = 0 \\) has a repeated real solution? Provide the steps to find the value of \\( k \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-b13fa5b43d87d7272ff5fa8c001b0df3904d18a1563730758763e5adf536b3c9 | What is the value of \( k \), if \( a-b \), \( a \), and \( a+b \) are the zeroes of the cubic polynomial \( kx^3 - 8x^2 + 5 \)? | [
"\\( k = \\pm\\frac{32}{3\\sqrt{15}} \\)"
] | [
"What is the value of \\( k \\), if \\( a-b \\), \\( a \\), and \\( a+b \\) are the zeroes of the cubic polynomial \\( kx^3 - 8x^2 + 5 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 7,
"tir": 2
} | 11 | 0 | 0 |
omr-ws-36933c2cecb276b546f3262364469a12a00e8de4e974cbd4740174d51c0dff75 | What is the value of \( m \) if both \( m \) and \( n \) are integers and \( x^2 - x - 1 \) is a factor of \( mx^7 + nx^6 + 1 \)? | [
"\\( m = 8 \\)"
] | [
"What is the value of \\( m \\) if both \\( m \\) and \\( n \\) are integers and \\( x^2 - x - 1 \\) is a factor of \\( mx^7 + nx^6 + 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-a6462980c5c1d55b5537df08f2555b6f41eb6bdf2f35d718afc6ced6cb502177 | What is the value of \( v_p (a^{p-1} - 1) \) where \( p \) is a prime and \( \gcd(a, p) = 1 \)? Is it always equal to 1? If not, provide a counterexample. | [
"2"
] | [
"What is the value of \\( v_p (a^{p-1} - 1) \\) where \\( p \\) is a prime and \\( \\gcd(a, p) = 1 \\)? Is it always equal to 1? If not, provide a counterexample."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-3183d1b6d4ba15aa2d6c6c35b06118b69cd9483de1189a77f46db9aeb403966d | What is the value of \( w^w \) where \( w^3 = 1 \)? | [
"1"
] | [
"What is the value of \\( w^w \\) where \\( w^3 = 1 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 1
} | 13 | 0 | 0 |
omr-ws-5835673ee880570c3758960c8c6f6e19cd4e375c47ecf75b036013f5858ed4a1 | What is the value of \( x \) if \( \arcsin(x) = \arccos\left(\frac{4}{5}\right) - \arccos\left(\frac{-4}{5}\right) \)? | [
"-\\dfrac{24}{25}"
] | [
"What is the value of \\( x \\) if \\( \\arcsin(x) = \\arccos\\left(\\frac{4}{5}\\right) - \\arccos\\left(\\frac{-4}{5}\\right) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e5b2a8aa9165ac92e5f46e1142ce6d5964be15ba982f94120c2d78b8e3715e51 | What is the value of \( x \) if \( x \) is a real number and \( x^2 - \lfloor x \rfloor^2 = 2014 \)? | [
"\\( \\sqrt{1009 \\cdot 1007} \\)"
] | [
"What is the value of \\( x \\) if \\( x \\) is a real number and \\( x^2 - \\lfloor x \\rfloor^2 = 2014 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 30
} | 52 | 0 | 0 |
omr-ws-ffa921984bb981ee92e25092f0a2486ce52ac206ad5396ad2d787ce7fb7fe158 | What is the value of \( x \) in the congruence \( 9^{2010} \equiv x \pmod{17} \)? | [
"13"
] | [
"What is the value of \\( x \\) in the congruence \\( 9^{2010} \\equiv x \\pmod{17} \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-54c8557d8222abd64f590a28a1a973a8e16d30376cc878c103ad9af349fc45ea | What is the value of \( x \) in the equation \( \frac{97x}{83} = \frac{727}{863} \)? | [
"\\( \\frac{60341}{83711} \\)"
] | [
"What is the value of \\( x \\) in the equation \\( \\frac{97x}{83} = \\frac{727}{863} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-f2276fd8839de2d0a8789129d2c8713bab32846d85671cc984dd31aded7e01cc | What is the value of \( x \) in the equation \( x^2 - 9x + 30 = 2 \sqrt{x^2 - 9x + 45} \)? | [
"\\( x = 5 \\) or \\( x = 4 \\)"
] | [
"What is the value of \\( x \\) in the equation \\( x^2 - 9x + 30 = 2 \\sqrt{x^2 - 9x + 45} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 18
} | 19 | 0 | 0 |
omr-ws-1f3df7af6be6ace345f2072204740a7ad81fb6b6af7106b9963e9490bf4e6ea3 | What is the value of \( x \) in the system of equations given by \[ \begin{cases} x + y + z = a \\ x^2 + y^2 + z^2 = b^2 \\ x^3 + y^3 + z^3 = a^3 \end{cases} \]? | [
"\\( a \\), \\( \\sqrt{\\frac{b^2-a^2}{2}} \\), or \\( -\\sqrt{\\frac{b^2-a^2}{2}} \\)"
] | [
"What is the value of \\( x \\) in the system of equations given by\n\\[\n\\begin{cases}\nx + y + z = a \\\\\nx^2 + y^2 + z^2 = b^2 \\\\\nx^3 + y^3 + z^3 = a^3\n\\end{cases}\n\\]?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-7354acdfc835f222137d8063f89041462d829337e276db2574ad363d7e29bf65 | What is the value of \((1 - w^2 + w)(1 + w^2 - w)\), where \(1, w, w^2\) are the cube roots of unity? | [
"4"
] | [
"What is the value of \\((1 - w^2 + w)(1 + w^2 - w)\\), where \\(1, w, w^2\\) are the cube roots of unity?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-28b56cccd43fbb6cf5de89af063f4dbbe31de0e5b47231e06db7b1b6160d0d95 | What is the value of \((3x - 2)(4x + 1) - (3x - 2)4x + 1\) when \(x = 4\)? | [
"11"
] | [
"What is the value of \\((3x - 2)(4x + 1) - (3x - 2)4x + 1\\) when \\(x = 4\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-161a4f53ae2c3a8709c7bb691ee09bac34df6930e746501f95864408161c9adb | What is the value of \((3x-1)(4x+1)-(3x-2)4x+1\) when \(x=4\)? | [
"28"
] | [
"What is the value of \\((3x-1)(4x+1)-(3x-2)4x+1\\) when \\(x=4\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-65071a03e732017fedfa1a7a10d19dea4d4508f80732222370a47d1dfccd5574 | What is the value of \((3x-2)(4x+1)-(3x-2)4x+1\) when \(x=4\)? | [
"11"
] | [
"What is the value of \\((3x-2)(4x+1)-(3x-2)4x+1\\) when \\(x=4\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-017fc906f224404a3024dd868a4d95fb1cd73f710f6ad88a07cf88b610754435 | What is the value of \((k^2 - 1^2)(k^2 - 2^2)(k^2 - 3^2) \cdots (k^2 - t^2) \mod p\) for any value of \(t < k\) and \(p\) prime? | [
"0"
] | [
"What is the value of \\((k^2 - 1^2)(k^2 - 2^2)(k^2 - 3^2) \\cdots (k^2 - t^2) \\mod p\\) for any value of \\(t < k\\) and \\(p\\) prime?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3ffb97f5f3fb121cc37572c70d887d2c80addb389af56e9301a5e2bd13a8faf1 | What is the value of \(-1\) in \(p\)-adic numbers? What is the value of \(\frac{1}{2}\) in \(3\)-adic numbers? | [
"\\overline{1}2"
] | [
"What is the value of \\(-1\\) in \\(p\\)-adic numbers? What is the value of \\(\\frac{1}{2}\\) in \\(3\\)-adic numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-ae1294ca3633ebe9aded78a79071878406cd56daecebe9f441a5a879e5fefb68 | What is the value of \(1 \times 3 \times 5 \times 7 \times \cdots \times 97 \times 99 \mod 100\)? | [
"75"
] | [
"What is the value of \\(1 \\times 3 \\times 5 \\times 7 \\times \\cdots \\times 97 \\times 99 \\mod 100\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-25a5dc8936445ce5c835801687702483562a0b72babc5cc866365d7e5ac13d36 | What is the value of \(199^{13^{15^{17}}} \mod 19\)? | [
"9"
] | [
"What is the value of \\(199^{13^{15^{17}}} \\mod 19\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ec5b680a4b048403f8772ac2e6f364cfe17a28130c2b445d8d4a2871de9b9d5 | What is the value of \(2009^{2009} \mod 1000\)? | [
"489"
] | [
"What is the value of \\(2009^{2009} \\mod 1000\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-411e31d1ee63140052b197cc661b40ab6041ecea85aedd0a14430b696ead3f6e | What is the value of \(2016^{2015^{2014^{\cdots^{3^{2^1}}}}}\) modulo 20? | [
"16"
] | [
"What is the value of \\(2016^{2015^{2014^{\\cdots^{3^{2^1}}}}}\\) modulo 20?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-eb7b9db0c3078985d2795f0b920e6f8c086b298fb3d792687081e156666aec44 | What is the value of \(255 \times 255 \times 255\)? | [
"16581375"
] | [
"What is the value of \\(255 \\times 255 \\times 255\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7344985c260a37111016a7706eb9d00d0933e78ce3e14beaa6328ac55d63cec2 | What is the value of \(2^{1990} \mod 1990\)? | [
"1024"
] | [
"What is the value of \\(2^{1990} \\mod 1990\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-8345a6e9b9ef2257607812604db83612225f67cf058fb038dce1209d0ea253af | What is the value of \(2^{n-1} \sin\left(\frac{1}{n}\pi\right) \sin\left(\frac{2}{n}\pi\right) \cdots \sin\left(\frac{n-1}{n}\pi\right)\)? | [
"n"
] | [
"What is the value of \\(2^{n-1} \\sin\\left(\\frac{1}{n}\\pi\\right) \\sin\\left(\\frac{2}{n}\\pi\\right) \\cdots \\sin\\left(\\frac{n-1}{n}\\pi\\right)\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4a35aaa5b87bccbf4c5678ceec221314e94921f30445f3cea05a91b0cd3cc57a | What is the value of \(3 + \frac{1}{3+\frac{1}{3+\frac{1}{3}}}\)? | [
"\\(\\frac{109}{33}\\)"
] | [
"What is the value of \\(3 + \\frac{1}{3+\\frac{1}{3+\\frac{1}{3}}}\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-849b7933dca2a72a75c6fd8423a1d97359b353fa53a8e3666d977ccd622f232c | What is the value of \(3^{2017} \mod 10000\)? | [
"163"
] | [
"What is the value of \\(3^{2017} \\mod 10000\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-87c92a8a9d7d4ced7e28d36309542268cece286aef09b097741e19c418825e35 | What is the value of \(8^{\log_4 6}\)? | [
"\\(6\\sqrt{6}\\)",
"\\(6^{\\frac{3}{2}}\\)"
] | [
"What is the value of \\(8^{\\log_4 6}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 52
} | 83 | 0 | 0 |
omr-ws-6144e892558fc7a6ff4654c89f241471cc80178ab44d6fdb0d00150e8bd59d04 | What is the value of \(\bigg(\Big((2+1)^{-1}+1\Big)^{-1}+1\bigg)^{-1}+1\)? | [
"\\(\\frac{11}{7}\\)"
] | [
"What is the value of \\(\\bigg(\\Big((2+1)^{-1}+1\\Big)^{-1}+1\\bigg)^{-1}+1\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-6c81de04147cb711eb605d8f7391ebc31b927ca16b5d0530f6f9d67c58ae742c | What is the value of \(\cos \frac{\pi}{6} + \cos \frac{2\pi}{6} + \cdots + \cos \frac{2022\pi}{6}\)? | [
"-1"
] | [
"What is the value of \\(\\cos \\frac{\\pi}{6} + \\cos \\frac{2\\pi}{6} + \\cdots + \\cos \\frac{2022\\pi}{6}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-a861f1384f31ca66ddeea8345894dc9f75a80fe9e6b003f53824106ffa7ce2e2 | What is the value of \(\frac{(1+\frac{1}{3})(1+\frac{1}{5})(1+\frac{1}{7})}{\sqrt{(1-\frac{1}{3^2})(1-\frac{1}{5^2})(1-\frac{1}{7^2})}}\)? | [
"2"
] | [
"What is the value of \\(\\frac{(1+\\frac{1}{3})(1+\\frac{1}{5})(1+\\frac{1}{7})}{\\sqrt{(1-\\frac{1}{3^2})(1-\\frac{1}{5^2})(1-\\frac{1}{7^2})}}\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2fcae45a76c214ca68ae13aa4fb06bb921a1799c049fe4303e7ba4ba8aa4bbaf | What is the value of \(\frac{(a^2 + b^2 + 2a^3 + 2b^4)^2}{(a+b)^2}\), if \(a^2 = \log_3 27 + \log_3 9\) and \(b^2 = \log_6 36 + \log_6 6\)? Express your answer in simplest form. | [
"\\(2352-588\\sqrt{15}+1040\\sqrt{5}-1300\\sqrt{3}\\)"
] | [
"What is the value of \\(\\frac{(a^2 + b^2 + 2a^3 + 2b^4)^2}{(a+b)^2}\\), if \\(a^2 = \\log_3 27 + \\log_3 9\\) and \\(b^2 = \\log_6 36 + \\log_6 6\\)? Express your answer in simplest form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 6,
"tir": 20
} | 32 | 0 | 0 |
omr-ws-769ef91543e2c7ad4425398559fb16070b07c364450d88bd8b7152ea8b93551a | What is the value of \(\frac{\cos 10^\circ}{\cos 5^\circ}\)? | [
"2\\cos 5^\\circ - \\sec 5^\\circ"
] | [
"What is the value of \\(\\frac{\\cos 10^\\circ}{\\cos 5^\\circ}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-0edc6ebd7277b0b5b8eba407e124b447612cba477fff51e78b4c5b53b8261af6 | What is the value of \(\frac{d(z^*)}{d(z^*)}\), where \(z^*\) is the complex conjugate of \(z\)? | [
"1"
] | [
"What is the value of \\(\\frac{d(z^*)}{d(z^*)}\\), where \\(z^*\\) is the complex conjugate of \\(z\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eea56a68fb69f71c3516b82a6d5acc8a2275fad38081c05f0e77ff8c8fb210f1 | What is the value of \(\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right) \cdot \left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)\)? | [
"21000"
] | [
"What is the value of \\(\\left(\\sum_{k=1}^{20} \\log_{5^k} 3^{k^2}\\right) \\cdot \\left(\\sum_{k=1}^{100} \\log_{9^k} 25^k\\right)\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7eb0acb72b29ff84ccf0a83310adcfd150e4c3e34a0f2d7ffa41ddbe959b4412 | What is the value of \(\left[2207 - \frac{1}{2207 - \frac{1}{2207 - \frac{1}{2207 - \cdots}}}\right]^{\frac{1}{8}}\)? | [
"\\dfrac{3 + \\sqrt{5}}{2}"
] | [
"What is the value of \\(\\left[2207 - \\frac{1}{2207 - \\frac{1}{2207 - \\frac{1}{2207 - \\cdots}}}\\right]^{\\frac{1}{8}}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-dd92686cfa6a12c65aca77ddc74ef7d3c743e9c52f199b0e7b0d1719211514cf | What is the value of \(\left\lfloor 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{2018}} \right\rfloor\)? | [
"88"
] | [
"What is the value of \\(\\left\\lfloor 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\cdots + \\frac{1}{\\sqrt{2018}} \\right\\rfloor\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-8eb54dae8281f0e563cdf5ded0a43ba56d0bf4ac666d172c4a0156f1f007f007 | What is the value of \(\left\lfloor \frac{a}{b} \right\rfloor + \left\lfloor \frac{2a}{b} \right\rfloor + \cdots + \left\lfloor \frac{(b-1)a}{b} \right\rfloor\), where \(\left\lfloor x \right\rfloor\) represents the greatest integer function? | [
"\\(\\frac{(a-1)(b-1) + \\gcd(a,b) - 1}{2}\\)"
] | [
"What is the value of \\(\\left\\lfloor \\frac{a}{b} \\right\\rfloor + \\left\\lfloor \\frac{2a}{b} \\right\\rfloor + \\cdots + \\left\\lfloor \\frac{(b-1)a}{b} \\right\\rfloor\\), where \\(\\left\\lfloor x \\right\\rfloor\\) represents the greatest integer function?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-853ff8bf222ecddfd852b005e60788b1f1cc7d4592495d2bc2807c1e85373478 | What is the value of \(\lim_{n \to \infty} \frac{\prod_{i=1}^{2n}(n^2+i^2)^{\frac{1}{n}}}{n^4}\)? | [
"\\( e^{\\int_0^2 \\log(1+x^2) \\, dx} \\)"
] | [
"What is the value of \\(\\lim_{n \\to \\infty} \\frac{\\prod_{i=1}^{2n}(n^2+i^2)^{\\frac{1}{n}}}{n^4}\\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-495030799cac88e2b301dee539c82d95479d574964593821f5248dfc8cd0233c | What is the value of \(\lim_{x \to 0} \ln(a)\)? Explain the reason. | [
"\\(\\ln(a)\\)"
] | [
"What is the value of \\(\\lim_{x \\to 0} \\ln(a)\\)? Explain the reason."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-5e38409070a6873326e63f6366d35817a8025609ebc97818dd7b4beb09c95275 | What is the value of \(\sin\pi \cosh2\pi + i\cos\pi \sinh2\pi\)? | [
"\\(i\\cdot\\frac{e^{-2\\pi}-e^{2\\pi}}2\\)"
] | [
"What is the value of \\(\\sin\\pi \\cosh2\\pi + i\\cos\\pi \\sinh2\\pi\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-1c3e61fe7c917143175d1b17a52f72ae23b6a9f6fa33cca75496617942246cca | What is the value of \(\tan 11^\circ + \tan 13^\circ + \tan 11^\circ \tan 34^\circ + \tan 32^\circ + \tan 34^\circ + \tan 32^\circ \tan 13^\circ\)? | [
"2"
] | [
"What is the value of \\(\\tan 11^\\circ + \\tan 13^\\circ + \\tan 11^\\circ \\tan 34^\\circ + \\tan 32^\\circ + \\tan 34^\\circ + \\tan 32^\\circ \\tan 13^\\circ\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dac298a5cccd4e244bf481d6a0d5b09855e2a4e737a54b318687d94580e7204d | What is the value of \(\tan(3\pi/2)\)? Is it \(+\infty\) or \(-\infty\)? Given that the period of \(\tan(x)\) is \(\pi\), if \(\tan(\pi/2) = +\infty\), should \(\tan(3\pi/2)\) also be \(+\infty\)? | [
"undefined"
] | [
"What is the value of \\(\\tan(3\\pi/2)\\)? Is it \\(+\\infty\\) or \\(-\\infty\\)? Given that the period of \\(\\tan(x)\\) is \\(\\pi\\), if \\(\\tan(\\pi/2) = +\\infty\\), should \\(\\tan(3\\pi/2)\\) also be \\(+\\infty\\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-7cf45dd99c9a24cf5182cab88670b358944a93c8a62ad27f1287d1c03c5562f5 | What is the value of \[ \log_37 \cdot \log_59 \cdot \log_711 \cdot \log_913 \cdots \log_{21}25 \cdot \log_{23}27? \] | [
"6"
] | [
"What is the value of \n\n\\[ \\log_37 \\cdot \\log_59 \\cdot \\log_711 \\cdot \\log_913 \\cdots \\log_{21}25 \\cdot \\log_{23}27? \\]"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-7108f3169c0f3faee983d6c85fd3b71faa30218c06602a806c85bc99da68bca1 | What is the value of \[ {100 \choose 0} + {100 \choose 4} + {100 \choose 8} + \cdots + {100 \choose 100}? \] | [
"\\(2^{98}-2^{49}\\)"
] | [
"What is the value of\n\\[\n{100 \\choose 0} + {100 \\choose 4} + {100 \\choose 8} + \\cdots + {100 \\choose 100}?\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-a9982a433b2eea5d5b56ee02e3b1b5a3fad50202b0bca118d17a414cd0047171 | What is the value of \[\bigg(\Big((2+1)^{-1}+1\Big)^{-1}+1\bigg)^{-1}+1?\] | [
"\\(\\frac{11}{7}\\)"
] | [
"What is the value of \\[\\bigg(\\Big((2+1)^{-1}+1\\Big)^{-1}+1\\bigg)^{-1}+1?\\]"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e9e6c92ab26f7238fc68ae5c7556b63bf16e55d397a967194bf176b449d1e170 | What is the value of \[\frac{x^1\cdot x^2\cdot x^3\cdots x^{15}}{x^2\cdot x^4 \cdot x^6 \cdots x^{20}}\]if $x=2$? | [
"1024"
] | [
"What is the value of \\[\\frac{x^1\\cdot x^2\\cdot x^3\\cdots x^{15}}{x^2\\cdot x^4 \\cdot x^6 \\cdots x^{20}}\\]if $x=2$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2d94c578585cb2312b99f1a1a4fe5973df335c16ca0a08fe4b4404ade04ebae9 | What is the value of \[\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16} + \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16}+\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16}+\tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?\] | [
"68"
] | [
"What is the value of \\[\\tan^2 \\frac {\\pi}{16} \\cdot \\tan^2 \\frac {3\\pi}{16} + \\tan^2 \\frac {\\pi}{16} \\cdot \\tan^2 \\frac {5\\pi}{16}+\\tan^2 \\frac {3\\pi}{16} \\cdot \\tan^2 \\frac {7\\pi}{16}+\\tan^2 \\frac {5\\pi}{16} \\cdot \\tan^2 \\frac {7\\pi}{16}?\\]"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 23
} | 44 | 0 | 0 |
omr-ws-a25dd4def9e48fa2fcafceba13b4d5db8fbb0971f0f5dd3370cab272cb610852 | What is the value of iterated tetration $^{^{^{2}}}^{^{2}}^{2}2$? | [
"^{65536}2"
] | [
"What is the value of iterated tetration $^{^{^{2}}}^{^{2}}^{2}2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-05d77744a0a3b46df977a8205d3090424ac96b2d9efc1b907e1b1a1b0fc7d4a4 | What is the value of the 25th term of the arithmetic sequence $2, 5, 8, \ldots$? | [
"74"
] | [
"What is the value of the 25th term of the arithmetic sequence $2,\n5, 8, \\ldots$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-16e915e1416de6adf847d9125b7d9942aca1b0f4e8c5853c0a1f1e6177bb6241 | What is the value of the Riemann zeta function at \( s = -4 \)? | [
"0"
] | [
"What is the value of the Riemann zeta function at \\( s = -4 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-6111f235d53959a4d9b434c0a3e74b9c01858251c03d569ade4131a7a196515d | What is the value of the continued fraction $3-\frac{2}{3-\frac{2}{3-\frac{2}{3- \cdots}}}$? | [
"2"
] | [
"What is the value of the continued fraction $3-\\frac{2}{3-\\frac{2}{3-\\frac{2}{3- \\cdots}}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d46b19afeba192a8ed667e705878ff18186e6c3740df35d34f0219927378629 | What is the value of the continued fraction $[1;2,3,4,5,6,7,8,9,10,\ldots]$? | [
"\\(\\frac{I_0(2)}{I_1(2)}\\)"
] | [
"What is the value of the continued fraction $[1;2,3,4,5,6,7,8,9,10,\\ldots]$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-712f8f4d71722dcd0fbd10f40e3d06b9bdf45152a2be1571bb6757b1f9f15e46 | What is the value of the continued fraction $[1;3,5,7,9,11,\ldots]$? | [
"\\dfrac{e^2 + 1}{e^2 - 1}"
] | [
"What is the value of the continued fraction $[1;3,5,7,9,11,\\ldots]$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-536d55793eb2671b4d01836be2ba29a904bda650ee71527ce19e29f22155d184 | What is the value of the determinant of the matrix: \[ \left[\begin{array}{cccc} C_{n}^{p+n} & C_{n}^{p+n+1} & \dots & C_{n}^{p+2n} \\ C_{n}^{p+n+1} & C_{n}^{p+n+2} & \dots & C_{n}^{p+2n+1} \\ \vdots & \vdots & \dots & \vdots \\ C_{n}^{p+2n} & C_{n}^{p+2n+1} & \dots & C_{n}^{p+3n} \end{array}\right] \] | [
"(-1)^{\\frac{n(n+1)}{2}}"
] | [
"What is the value of the determinant of the matrix:\n\\[ \\left[\\begin{array}{cccc} C_{n}^{p+n} & C_{n}^{p+n+1} & \\dots & C_{n}^{p+2n} \\\\ C_{n}^{p+n+1} & C_{n}^{p+n+2} & \\dots & C_{n}^{p+2n+1} \\\\ \\vdots & \\vdots & \\dots & \\vdots \\\\ C_{n}^{p+2n} & C_{n}^{p+2n+1} & \\dots & C_{n}^{p+3n} \\end{array}\\ri... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-48e69321c60d52c30f4500a31f0f9359a4e426d0e4cee8c0bafd6b8b1fe8e17c | What is the value of the expression $(25 + 8)^2 - (8^2 +25^2)$? | [
"400"
] | [
"What is the value of the expression $(25 + 8)^2 - (8^2 +25^2)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3724226c8c21180fa7c226f4526b1aa11405ba112480fa7525df56aa03f38c83 | What is the value of the expression $(37 + 12)^2 - (37^2 +12^2)$? | [
"888"
] | [
"What is the value of the expression $(37 + 12)^2 - (37^2 +12^2)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b13ae912472888008f694e9b68b00d273ce34b7091d1af6cd9fbeafba1c043b | What is the value of the expression $1 \cdot 2008^{99} + 2 \cdot 2008^{98} + 3 \cdot 2008^{97} + \cdots + 99 \cdot 2008^1 + 100 \cdot 2008^0$? | [
"\\(\\frac{2008^{101} - 202708}{2007^2}\\)"
] | [
"What is the value of the expression $1 \\cdot 2008^{99} + 2 \\cdot 2008^{98} + 3 \\cdot 2008^{97} + \\cdots + 99 \\cdot 2008^1 + 100 \\cdot 2008^0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 7
} | 24 | 0 | 0 |
omr-ws-c8f8b7b012aef44d06194f72a1ee0aa11553b969ff7f84ca4c070dd81c0d2790 | What is the value of the expression $3 + \sqrt {3^2 + \sqrt {3^4 + \sqrt {3^8 + \sqrt {...}}}}$? | [
"\\(\\frac{9 + 3\\sqrt{5}}{2}\\)"
] | [
"What is the value of the expression $3 + \\sqrt {3^2 + \\sqrt {3^4 + \\sqrt {3^8 + \\sqrt {...}}}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 15
} | 37 | 0 | 0 |
omr-ws-3a7b1d6f4fac25c376372a04a50590a19cdcc651dfe0ab1b549f77332a34b032 | What is the value of the expression $3^{\sqrt{\log_{3}2}}-2^{\sqrt{\log_{2}3}}$? | [
"0"
] | [
"What is the value of the expression $3^{\\sqrt{\\log_{3}2}}-2^{\\sqrt{\\log_{2}3}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 12
} | 29 | 0 | 0 |
omr-ws-cfa3147db9f2c6a2a1389370c1eb62b3a4f652586ae34da7f7362e8ec9a08307 | What is the value of the expression $5^3 - 3 \times 5^2 + 3 \times 5 - 1$? | [
"64"
] | [
"What is the value of the expression $5^3 - 3 \\times 5^2 + 3 \\times 5 - 1$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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